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\title{The Limit Superior and Inferior of Real Sequence}
\author{JIAQI TANG}
\date{\today}%直接改July 5, 2025亦可.
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\begin{document}
\maketitle
\begin{abstract}
	This is a lecture script for the Real Analysis course, whose content mainly introduces the limit superior and limit inferior of real sequences.
\end{abstract}
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\section{Preliminary Knowledge}
Before formally introducing the limit superior and limit inferior of sequences, we need to know some preliminary content.
\begin{theorem}
	A monotonic sequence must have a limit (allowing $\pm \infty$).
\end{theorem}
\begin{proof}
	If a monotonic sequence is bounded, it falls under the theorem in the general case (i.e., not allowing $\pm \infty$). If it is unbounded, then the limit of a monotonically increasing sequence is defined as $+\infty$; the limit of a monotonically decreasing sequence is defined as $-\infty$.
\end{proof}
\begin{theorem}
	Any sequence must have a convergent subsequence (allowing convergence to $\pm \infty$).
\end{theorem}
\begin{proof}
	If the sequence is bounded, then the theorem in the general case tells us that we can always extract a convergent subsequence from it. If it is unbounded, then for each $k\in\mathbb{N}$, there must be infinitely many $x_i$ in $\mathbb{R}-(-k,k)$. Then we can certainly choose infinitely many of them such that they all lie in $(k,+\infty)$ or $(-\infty,-k)$. This is the subsequence that converges to $+\infty$ or $-\infty$ that we are looking for.
\end{proof}
The above two theorems are natural results of extending the concept of convergence to allow limits to be $\pm\infty$.

\section{Definition of Limit Superior and Limit Inferior of Real Sequences}
We now discuss the limit superior and limit inferior of real sequences. As the concept of convergence is extended to allow limits to be $\pm\infty$, naturally, the supremum and infimum of number sets can also be extended to allow $\pm \infty$.

Here, we say that the supremum (resp. infimum) of a number set $A$ is $\pm\infty$ refers to the naturally extended definition, i.e., if there is a subsequence in $A$ that converges to $+\infty$ (resp. $-\infty$), then its supremum (resp. infimum) is defined as $+\infty$ (resp. $-\infty$). If no such subsequence can be extracted, then the definition of the supremum/infimum remains the same as before.

\begin{lemma}[A Characterization]\label{lemma}
	Let $\{x_n\}_{n\in\mathbb{N}}$ be a sequence of real numbers, then
	\begin{align}
		\label{1.5.6}\sup_{n\in \mathbb{N}}x_n=&\lim_{m\to\infty}\max\{x_1,x_2,\dots,x_m\},\\
		\label{1.5.7}\inf_{n\in \mathbb{N}}x_n=&\lim_{m\to\infty}\min\{x_1,x_2,\dots,x_m\}.
	\end{align}
\end{lemma}
This lemma provides a characterization of the supremum and infimum. The proof of \eqref{1.5.6} simply notes that $\big\{\max\{x_1,x_2,\dots,x_m\}\big\}_{m\in\mathbb{N}}$ is a monotonically increasing sequence, and \eqref{1.5.7} is similar.

For a general real sequence, although the limit may not exist, we can always define its \textbf{limit superior} and \textbf{limit inferior}:
\begin{definition}[Definition of Limit Superior and Limit Inferior]
	For any given real sequence $\{x_n\}_{n\geqslant 1}$, for any $n\geqslant 1$, we let
	\begin{align*}
		\overline{x}_n=\sup_{\ell\geqslant n}x_\ell,\quad \underline{x}_n=\inf_{\ell\geqslant n}x_\ell.
	\end{align*}
	Clearly, $\{\overline{x}_n\}_{n\geqslant 1}$ is a monotonically decreasing sequence, and $\{\underline{x}_n\}_{n\geqslant 1}$ is a monotonically increasing sequence. Therefore, $\{\overline{x}_n\}_{n\geqslant 1}$ and $\{\underline{x}_n\}_{n\geqslant 1}$ both have limits (which can be infinite). Accordingly, we define the limit superior and limit inferior of the sequence $\{x_n\}_{n\geqslant 1}$ as
	\begin{align}
		\label{def1}\varlimsup_{n\to \infty} x_n=&\limsup_{n\to \infty} x_n=\lim_{n\to \infty}\overline{x}_n=\lim_{n\to\infty}\left(\sup_{\ell\geqslant n}x_{\ell}\right),\\
		\label{def2}\varliminf_{n\to \infty} x_n=&\liminf_{n\to \infty} x_n=\lim_{n\to \infty}\underline{x}_n=\lim_{n\to\infty}\left(\inf_{\ell\geqslant n}x_{\ell}\right).
	\end{align}
\end{definition}

\begin{rmk}
	Another definition is that the maximum (resp. minimum) value among the limits of all convergent subsequences of a real sequence $\{x_n\}_{n\in\mathbb{N}}$ is called the limit superior (resp. limit inferior) of $\{x_n\}$, denoted in the same way as above.
\end{rmk}

\begin{rmk}[Terence Tao]
	Imagine a piston starting at $+\infty$ and moving leftward along the real number line until it encounters an element of the sequence $a_1,a_2,\dots$ and stops. The position where it stops is the supremum of $a_1,a_2,\dots $, which we denote as $a_1^+$. Now we remove the first term $a_1$ from the sequence; this may cause our piston to slide left and stop at a new position $a_2^+$ (obviously, in many cases, the piston remains in place, and $a_2^+$ is exactly the same as $a_1^+$). Then we remove the second term $a_2$, causing the piston to slide left a bit more. If we continue this process, the piston keeps sliding left. Eventually, there is a position where the piston stops and no longer slides left. This position is the limit superior of the sequence. A similar analogy can describe the limit inferior of the sequence.
\end{rmk}
According to the definition, we directly have the following:
\begin{theorem}
	\begin{align*}
		\inf_{n\in\mathbb{N}} x_n\leqslant \varliminf_{n\to\infty}x_n\leqslant \varlimsup_{n\to\infty}x_n\leqslant\sup_{n\in\mathbb{N}}x_n
	\end{align*}
\end{theorem}

\begin{example}
	Let's look at some examples, which can help us understand the limit superior and limit inferior.
	\begin{enumerate}
		\item For $\left\{1,1/2,3,1/4,\dots,n^{(-1)^{n-1}},\dots\right\}$, $\varlimsup\limits_{n\to\infty}x_n=+\infty$, $\varliminf\limits_{n\to\infty}x_n=0$.
		\item For $\left\{1,-1,2,-2,\dots,n,-n,\dots\right\}$, $\varlimsup\limits_{n\to\infty}x_n=+\infty$, $\varliminf\limits_{n\to\infty}x_n=-\infty$.
		\item For $\left\{1!,2!,3!,\dots,n!,\dots\right\}$, $\varlimsup\limits_{n\to\infty}x_n=\varliminf\limits_{n\to\infty}x_n=+\infty$.
		\item For $\left\{1,2^{-1},\dots,n^{-1},\dots\right\}$, $\lim\limits_{n\to\infty}x_n=\varlimsup\limits_{n\to\infty}x_n=\varliminf\limits_{n\to\infty}x_n=0$.
	\end{enumerate}
\end{example}

\section{Properties of Limit Superior and Limit Inferior of Real Sequences}
It is necessary to understand some basic properties of the limit superior and limit inferior of sequences, as this will greatly facilitate their use.
\begin{theorem}[Existence]
	The limit superior and limit inferior of any real sequence $\{x_n\}_{n\in\mathbb{N}}$ must exist, and we have
	\begin{align*}
		\varliminf_{n\to\infty}x_n=\lim_{n\to\infty}\lim_{m\to\infty}\min\{x_n,x_{n+1},\dots,x_{n+m}\},\\
		\varlimsup_{n\to\infty}x_n=\lim_{n\to\infty}\lim_{m\to\infty}\max\{x_n,x_{n+1},\dots,x_{n+m}\}.
	\end{align*}
\end{theorem}
\begin{proof}
	This follows directly from Lemma \ref{lemma}.
\end{proof}
\begin{theorem}
	A necessary and sufficient condition for $\{x_n\}$ to be a convergent sequence is
	\begin{align*}
		\varlimsup_{n\to\infty}x_n=\varliminf_{n\to\infty}x_n.
	\end{align*}
\end{theorem}
\begin{theorem}
	Let $\{y_n\}$ be a convergent sequence. When the right-hand side of the following equations has a definite meaning (i.e., does not involve indeterminate forms), we have
	\begin{align*}
		\varlimsup_{n\to\infty}(x_n+y_n)&=\varlimsup_{n\to\infty}x_n+\lim_{n\to\infty}y_n,\\
		\varliminf_{n\to\infty}(x_n+y_n)&=\varliminf_{n\to\infty}x_n+\lim_{n\to\infty}y_n.
	\end{align*}
\end{theorem}
\begin{theorem}
	$\{x_n\}_{n\in\mathbb{N}}$ and $\{y_n\}_{n\in\mathbb{N}}$ are two sequences. If $x_n\leqslant y_n(n=1,2,\dots)$, then
	\begin{align*}
		\varliminf_{n\to\infty}x_n\leqslant\varliminf_{n\to\infty}y_n,\quad\varlimsup_{n\to\infty}x_n\leqslant\varlimsup_{n\to\infty}y_n.
	\end{align*}
\end{theorem}
\begin{theorem}
	If $\alpha\in\mathbb{R}-\{0\}$ is a nonzero real number,
	\begin{align*}
		\varlimsup_{n\to\infty}\alpha x_n=\alpha\varlimsup_{n\to\infty}x_n,\quad\varliminf_{n\to\infty}\alpha x_n=\alpha\varliminf_{n\to\infty}x_n
	\end{align*}
	(Actually, the case where $\alpha$ is $0$ also holds, but it is too trivial.)
\end{theorem}

\section{Characterization of Limit Superior and Limit Inferior of Real Sequences}
First, supplement a few definitions:
\begin{itemize}
	\item $\alpha\in \mathbb{R}$ is called a \textbf{cluster point} of $\{a_n\}_{n\geqslant 1}$ if for any $\varepsilon>0$, there exist infinitely many $n$ such that $a_n\in(\alpha-\varepsilon,\alpha+\varepsilon)$;
	\item $+\infty$ is called a \textbf{cluster point} of $\{a_n\}_{n\geqslant 1}$ if for any $M>0$, there exist infinitely many $n$ such that $a_n\in(M,+\infty)$;
	\item $-\infty$ is called a \textbf{cluster point} of $\{a_n\}_{n\geqslant 1}$ if for any $M>0$, there exist infinitely many $n$ such that $a_n\in(-\infty,-M)$.
\end{itemize}
\begin{proposition}
	$\alpha\in \mathbb{R}$ is a cluster point of $\{a_n\}_{n\geqslant 1}$ if and only if the sequence has a subsequence $\{a_{n_k}\}_{k\geqslant 1}$ that converges to $\alpha$.
\end{proposition}
\begin{proposition}
	$+\infty$ is a cluster point of $\{a_n\}_{n\geqslant 1}$ if and only if the sequence has a subsequence $\{a_{n_k}\}_{k\geqslant 1}$ such that $\lim\limits_{k\to\infty}a_{n_k}=+\infty$.
\end{proposition}

Let $E=\big\{\alpha\in\mathbb{R}\big|\alpha\text{ is a cluster point of }\{a_n\}_{n\geqslant 1}\big\}$ be the set of all cluster points of $\{a_n\}_{n\geqslant 1}$. It is a subset of $\mathbb{R}\cup \{\pm\infty\}$. We have:
\begin{proposition}
	$E\neq\varnothing$.
\end{proposition}
\begin{proposition}
	$E\subseteq\mathbb{R}$ if and only if the sequence $\{a_n\}_{n\geqslant 1}$ is bounded.
\end{proposition}
\begin{proposition}
	Assume $\{a_n\}_{n\geqslant 1}$ is bounded, then
	\begin{align*}
		\sup E=\varlimsup_{n\to\infty}a_n,\quad\inf E=\varliminf_{n\to\infty}a_n.
	\end{align*}
\end{proposition}
\begin{proposition}
	Assume $\{a_n\}_{n\geqslant 1}$ is bounded. Let $a^*=\varlimsup\limits_{n\to\infty}a_n$, then
	\begin{enumerate}
		\item $a^*\in E$, so $\sup E\in E$;
		\item For any $x>a^*$, there exists $N\in\mathbb{Z}_{\geqslant 1}$ such that for any $n>N$, we have $a_n<x$.
	\end{enumerate}
\end{proposition}
\end{document}